© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n72 K11n74

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X20,10,21,9 X11,16,12,17 X13,6,14,7 X15,18,16,19 X17,12,18,13 X22,20,1,19 X10,22,11,21

Gauss Code:

{1, -4, 2, -1, -3, 7, 4, -2, 5, -11, -6, 9, -7, 3, -8, 6, -9, 8, 10, -5, 11, -10}

DT (Dowker-Thistlethwaite) Code:

4 8 -14 2 20 -16 -6 -18 -12 22 10

Alexander Polynomial:

t-2 - 2t-1 + 3 - 2t + t2

Conway Polynomial:

1 + 2z2 + z4

Other knots with the same Alexander/Conway Polynomial:

{820, 10140, K11n74, ...}

Determinant and Signature:

{9, 0}

Jones Polynomial:

- q-4 + q-3 - q-2 + q-1 + 2 - q + 2q2 - 3q3 + 2q4 - 2q5 + q6

Other knots (up to mirrors) with the same Jones Polynomial:

{K11n74, ...}

A2 (sl(3)) Invariant:

- q-12 - q-10 - 2q-8 + q-4 + 4q-2 + 5 + 2q2 + q4 - 3q6 - 2q8 - 2q10 - q12 + q14 + q18

HOMFLY-PT Polynomial:

2a-4 + 3a-4z2 + a-4z4 - 8a-2 - 12a-2z2 - 6a-2z4 - a-2z6 + 11 + 15z2 + 7z4 + z6 - 4a2 - 4a2z2 - a2z4

Kauffman Polynomial:

3a-6z2 - 4a-6z4 + a-6z6 - 4a-5z + 10a-5z3 - 9a-5z5 + 2a-5z7 + 2a-4 + a-4z2 - a-4z4 - 3a-4z6 + a-4z8 - 13a-3z + 24a-3z3 - 16a-3z5 + 3a-3z7 + 8a-2 - 13a-2z2 + 11a-2z4 - 6a-2z6 + a-2z8 - 17a-1z + 29a-1z3 - 15a-1z5 + 2a-1z7 + 11 - 19z2 + 18z4 - 8z6 + z8 - 13az + 25az3 - 14az5 + 2az7 + 4a2 - 8a2z2 + 10a2z4 - 6a2z6 + a2z8 - 5a3z + 10a3z3 - 6a3z5 + a3z7

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, -2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 1173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     1   j = 9                   1 1   j = 7                 2 1     j = 5             1 1 1       j = 3             1 2         j = 1         1 3 1           j = -1       1   2             j = -3       1 1               j = -5   1 1                   j = -7                         j = -9 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 73]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 73]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[11, 16, 12, 17], X[13, 6, 14, 7], X[15, 18, 16, 19], > X[17, 12, 18, 13], X[22, 20, 1, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 73]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -11, -6, 9, -7, 3, -8, 6, -9, 8, 10, > -5, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 73]]
Out[5]=	DTCode[4, 8, -14, 2, 20, -16, -6, -18, -12, 22, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 73]][t]
Out[6]=	-2 2 2 3 + t - - - 2 t + t t
In[7]:=	Conway[Knot[11, NonAlternating, 73]][z]
Out[7]=	2 4 1 + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73], > Knot[11, NonAlternating, 74]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 73]], KnotSignature[Knot[11, NonAlternating, 73]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 73]][q]
Out[10]=	-4 -3 -2 1 2 3 4 5 6 2 - q + q - q + - - q + 2 q - 3 q + 2 q - 2 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 73], Knot[11, NonAlternating, 74]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 73]][q]
Out[12]=	-12 -10 2 -4 4 2 4 6 8 10 12 14 5 - q - q - -- + q + -- + 2 q + q - 3 q - 2 q - 2 q - q + q + 8 2 q q 18 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 73]][a, z]
Out[13]=	2 2 4 4 2 8 2 2 3 z 12 z 2 2 4 z 6 z 11 + -- - -- - 4 a + 15 z + ---- - ----- - 4 a z + 7 z + -- - ---- - 4 2 4 2 4 2 a a a a a a 6 2 4 6 z > a z + z - -- 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 73]][a, z]
Out[14]=	2 2 2 8 2 4 z 13 z 17 z 3 2 3 z z 11 + -- + -- + 4 a - --- - ---- - ---- - 13 a z - 5 a z - 19 z + ---- + -- - 4 2 5 3 a 6 4 a a a a a a 2 3 3 3 13 z 2 2 10 z 24 z 29 z 3 3 3 4 > ----- - 8 a z + ----- + ----- + ----- + 25 a z + 10 a z + 18 z - 2 5 3 a a a a 4 4 4 5 5 5 4 z z 11 z 2 4 9 z 16 z 15 z 5 3 5 > ---- - -- + ----- + 10 a z - ---- - ----- - ----- - 14 a z - 6 a z - 6 4 2 5 3 a a a a a a 6 6 6 7 7 7 6 z 3 z 6 z 2 6 2 z 3 z 2 z 7 3 7 > 8 z + -- - ---- - ---- - 6 a z + ---- + ---- + ---- + 2 a z + a z + 6 4 2 5 3 a a a a a a 8 8 8 z z 2 8 > z + -- + -- + a z 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 73]], Vassiliev[3][Knot[11, NonAlternating, 73]]}
Out[15]=	{2, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 73]][q, t]
Out[16]=	2 1 1 1 1 1 1 q 3 5 - + 3 q + ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t + q t + q 9 5 5 4 5 3 3 2 2 3 t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 2 q t + q t + q t + 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n73

K11n72 K11n74